nxtgm is C++ 17 library for optimization of discrete graphical models. It is the inofficial successor of OpenGM

• Loopy Belief Propagation
• Dynamic Programming
• Brute Force
• Iterated Conditional Modes (ICM)
• Graph-Cut
• QPBO
• Higher order QPBO
• Integer Linear Programming (ILP)
• Hungarian Matching

Meta optimizers that use other optimizers

• Partial Optimality Reduced Optimization
• Chained Optimization
• Fusion Moves

• Higher order clique reduction:

  • Alexander Fix's higher order clique reduction

• LP / ILP Solvers:

  • Highs
  • Coin CLP / CBC

• Min-St-Cut / Maxflow:

  • Kolmogorov's maxflow algorithm

• QPBO:

  • Kolmogorov's famous QPBO algorithm wrapped in a plugin interface

• Proposal Generators:

  • α-expansion
  • optimizer based proposal generator (uses an optimizer like belief propagation generate proposals)

• Fusion Moves: resulting solution is not worse than the two original solutions

• Linux (x86-64)
• MacOS (x86-64, apple-silicon-arm64)
• Windows (x86_64)
• WebAssembly (emscripten-wasm-32)

nxtgm has language bindings for python and javascript. The python bindings are generated using pybind11 and the javascript bindings are generated using emscripten / embind.

Below is an example of how to use nxtgm in C++, Python and Javascript. The example shows how to create a discrete graphical model with 4 variables and 2 labels per variable. The unary factors are random and the pairwise factors are all the same potts function. We optimize the graphical model using loopy belief propagation. This example should illustrate that nxtgm has a homogeneous interface for all languages. Note that this example is not meant to be a good example for discrete graphical models. It is just meant to illustrate the interface.

auto num_var = 4;
auto num_labels = 2;
auto gm = nxtgm::DiscreteGm(4, 2);

// unary factors (with random values, just for the example)
for(auto vi = 0; vi < num_var; ++vi) {
    auto tensor = xt::random::rand<double>({n_labels}, 0.0, 1.0);
    auto func = std::make_unique<nxtgm::XArray>(tensor);
    auto func_index = gm.add_energy_function(std::move(f));
    gm.add_factor({vi}, func_index);
}

// pairwise factors along a chain, all with the same potts function
auto potts_func = std::make_unique<nxtgm::Potts>(num_labels, 0.5);
auto potts_func_index = gm.add_energy_function(std::move(potts_func));
for(auto vi0=0; vi0 < num_var - 1; ++vi0) {
    gm.add_factor({vi0, vi0+1}, potts_func_index);
}

// optimizer
auto optimizer_name = std::string("belief_propagation");
auto parameters = nxtgm::OptimizerParameters();
parameters["max_iterations"] = 10;
parameters["damping"] = 0.9;
parameters["convergence_tolerance"] = 1e-2;
auto optimizer = nxtgm::discrete_gm_optimizer_factory(gm, optimizer_name, parameters);

// optimize
auto status = optimizer.optimize();
auto solution = optimizer.best_solution();

num_var = 4
num_labels = 2
gm = nxtgm.DiscreteGm(num_var, num_labels)

# unary factors (with random values, just for the example)
for vi in range(num_var):
    unaries = np.random.rand(num_labels)
    func_index = gm.add_energy_function(func)
    gm.add_factor([vi], func_index)

# pairwise factors along a chain, all with the same potts function
potts_func = nxtgm.Potts(num_labels, 0.5)
potts_func_index = gm.add_energy_function(potts_func)
for vi0 in range(num_var - 1):
    gm.add_factor([vi0, vi0+1], potts_func_index)

# optimizer
optimizer_name = "belief_propagation"
parameters = dict(
    max_iterations=10,
    damping=0.9,
    convergence_tolerance=1e-2
)
optimizer = nxtgm.discrete_gm_optimizer_factory(gm, optimizer_name, parameters)

# optimize
status = optimizer.optimize()
solution = optimizer.best_solution()

let num_var = 4;
let num_labels = 2;
let gm = new nxtgm.DiscreteGm(num_var, num_labels);

// unary factors (with random values, just for the example)
for(let vi = 0; vi < num_var; ++vi) {
    let unaries = new Float64Array([
        Math.random(), Math.random()
    ]);
    let func = new nxtgm.XArray([num_labels], unaries);
    let func_index = gm.add_energy_function(func);
    gm.add_factor([vi], func_index);
}

// pairwise factors along a chain, all with the same potts function
let potts_func = new nxtgm.Potts(num_labels, 0.5);
let potts_func_index = gm.add_energy_function(potts_func);
for(let v0=0; v0 < num_var - 1; ++v0) {
    gm.add_factor([v0, v0+1], potts_func_index);
}

// optimizer
let optimizer_name = "belief_propagation";
let parameters = new nxtgm.OptimizerParameters();
parameters.set_int("max_iterations", 10);
parameters.set_double("damping", 0.9);
parameters.set_double("convergence_tolerance", 1e-2);
let optimizer = nxtgm.discrete_gm_optimizer_factory(gm, optimizer_name, parameters);

// optimize
let status = optimizer.optimize();
let solution = optimizer.best_solution();

• runtime polymorphism instead of template meta programming for the value-tables / tensors / functions. This makes it easier to generate language bindings for other languages and simplifies the packaging.

• plugin architecture for optimizers. This makes it easier to add new optimizers and simplifies the packaging. Furthermore optimizers can depend on other optimizers without relying on templates. This allows to generate language bindings for other languages.

• extensive testing. OpenGM had almost no tests beyond second order graphical models. nxtgm has tests for a broad range of graphical models.

nxtgm has a plugin architecture. That means each optimizer is a plugin. The plugins system we use is xplugin where each plugin is implemented as a shared / dynamic library.

This plugin-based architecture allows to easily add new optimizers to nxtgm and simplifies the packaging. Instead of depending on concrete implementations with potential bad licensing, we can just depend on the plugin interface.

Furthermore, optimizers can depend on other plugin interfaces. For example, the graph-cut optimizer uses on the min_st_cut plugins

We define multiple plugin interfaces:

• discrete_gm_optimizers: an interface for discrete graphical model optimizers (e.g. loopy belief propagation, icm, brute force)

• qpbo: an interface for QPBO-like optimizers. At the moment, there is only one implementation of this interface, which is QPBO from Vladimir Kolmogorov.

• min_st_cut: an interface for maxflow/min-st-cut-like optimizers. At the moment, there is only one implementation of this interface, which is maxflow from Vladimir Kolmogorov.

• horc: is an interface for higher order clique reduction. At the moment, there is only one implementation of this interface, which is horc from Alexander Fix.

• ilp: is an interface for integer linear programming solvers. At the moment, there are two implementations of this interface, which is highs and coin_clp.

• proposal_generators: is an interface for proposal generators. At the moment, there are two implementations of this interface, which is alpha_expansion and optimizer_based_proposal_generator. The optimizer_based_proposal_generator uses an optimizer like belief propagation to generate proposals.

The nxtgm library itself (ie the shared libraries and the header files) are licensed under the MIT license. See LICENSE_LIBRARY for more details. The plugins are licensed under different licenses. See LICENSE_PLUGINS for more details.

• Discrete graphical models used to be importat for computer vision. However, nowadays, deep learning is used for many tasks. Nevertheless, discrete graphical models are still important for some niche applications. For example, in the fields where extem little data / ground truth is available, discrete graphical models are still applicable. Or for extremly combinatorial problems,or highly constraint problems , ie matching problems, discrete graphical models are still very useful.

• Since ~2017 OpenGM is not maintained anymore and is beyond repairablity (I can say that, I am one of the main authors of OpenGM). The main reason for this is that OpenGM is written in C++03 and uses a lot of template meta programming. This makes is very hard to generate language bindings for other languages and package opengm properly.

• With the rise of tools like copilot, it was easy to port big parts of OpenGM to this library.

• I wanted to learn more about plugin architectures for C++ and felt discrete graphical models are an excellent use case for this.

• I want to give non-experts the possibility to use discrete graphical models.

• With emscripten it is possible to compile C++ code to WebAssembly. This allows to run discrete graphical models in the browser. This is very useful for teaching purposes.